An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

Eighth Chapter.
Calculation of functions by infinite series. General theorems
concerning series of powers.
37. We now proceed to employ the successive derived functions
of a given function in presenting the Theorem of the Mean Value in a
form which constitutes the basis of the mnost important theorem of the
Differential Calculus.
Let f(x) be a unique function from a to b, let its derived functions
f'(x), f"(x),... f(n-1)(x) be everywhere in the same interval continuous and therefore also finite, while we assume no other property of
the nth derived f(n)(x), but that it has the same value at each point
when formed progressively as regressively. Our first enquiry, in conformity with ~ 22, is whether the quotient
f(b) - f(a) - (b - ca f' (a)
(b - a)2
which again may be denoted by K, can be expressed by means of
higher derived functions. From the equation
f(b) - f(a) - (b - a) f' (a) -- K (b - a) = 0
it results as in ~ 22 that
qp(x) = f(b) - f(x) -- (b - x)f' (x) - (b -- x)2K
is a continuous function with a determinate differential quotient, and
that it vanishes for x -- a and for x = b. There must therefore be
some value x1, such that
p' (xl) = - f'(x) + f' (x) - (b - x) f"(x) + 2(b - x,) K = 0,
that is
K f"(x).
Accordingly we have the equation:
f(b) =f(a) + (b - a)f'(a) -+ i (b - a)2f"(a + 0 (b - a)), < 0 < 1.
If we proceed similarly and put
f(b) - f (a) - (b -- a)f (a) -j (b - a)2 f"(a) - (b - a)3 K -= 0,
the value of K is found by the equation:
HARNACK, Calculus. 5